Adam was in the thick of the action in the first half, opening the scoring on the half-hour mark before being involved in the incident that saw Wes Brown sent off nine minutes before half-time.
Brown was adjudged to have recklessly tackled the former Liverpool midfielder and was given his marching orders.
N'Zonzi doubled Stoke's lead nine minutes from the end as Mark Hughes' men made the most of their numerical advantage to record a first Premier League win since the end of August.
The result moves Stoke three points clear of the relegation places and drops Sunderland to bottom.
Hughes made one change to his Stoke starting line-up, with Glenn Whelan coming in for Wilson Palacios, while Sunderland manager Gus Poyet named an unchanged team after a 1-0 win over Manchester City prior to the international break.
It was Sunderland who were first to find their rhythm. Adam Johnson and Emanuele Giaccherini combined to set up Jack Colback, whose shot was blocked, before Stoke goalkeeper Asmir Begovic brilliantly denied Steven Fletcher from a one-on-one opportunity.
Despite the clever passing of Adam and some neat touches from Peter Crouch, Stoke struggled to create a meaningful chance until breaking the deadlock on the half-hour mark.
N'Zonzi raced on to Geoff Cameron's long punt forward, held it up and stroked the ball back to the onrushing Adam, who passed into the bottom corner from just inside the box for his third goal of the season.
Things got considerably worse for Sunderland on 35 minutes. Brown appeared to have made a fair sliding challenge, but referee Kevin Friend adjudged the 34-year-old defender had dangerously fouled Adam in the process and sent him off.
Stoke enjoyed much of the possession at the start of the second half.
However, both Jonathan Walters and Crouch failed to connect with an inviting Cameron cross, before Adam burst through the midfield but failed to keep his shot down.
Johnson found space on the edge of the opposition box on 70 minutes, but the winger, who scored a 25-yard strike for Manchester City against Stoke in the 2011-12 season, dragged his shot wide.
Begovic had to be alert to race from his line and, with an outstretched leg, clear the ball before Fletcher. The Bosnia-Herzegovina international caught the Scotland striker in the process, but Sunderland’s appeals for a penalty were waved away.
Sunderland were enjoying a good spell of pressure, but were hit on the counter-attack with nine minutes remaining. Adam fed Couch, who slid the ball between two defenders for N'Zonzi to clip it home.
The former France Under-21 international became Stoke's 10th scorer in the Premier League this season.
